We are solving for $t$ when $C(t) = 15$ using the equation:

$C(t) = 8 + 10e^{-0.03t}.$

Step 1: Set $C(t) = 15$

$15 = 8 + 10e^{-0.03t}.$

Step 2: Isolate $e^{-0.03t}$

$15 - 8 = 10e^{-0.03t},$ $7 = 10e^{-0.03t}.$ $e^{-0.03t} = \frac{7}{10} = 0.7.$

Step 3: Solve for $t$ using the natural logarithm

Take the natural logarithm of both sides: $\ln(e^{-0.03t}) = \ln(0.7).$ Using the property $\ln(e^x) = x$, this simplifies to: $-0.03t = \ln(0.7).$

Step 4: Calculate $t$

$t = \frac{\ln(0.7)}{-0.03}.$

First, calculate $\ln(0.7)$: $\ln(0.7) \approx -0.3567.$

Now divide: $t = \frac{-0.3567}{-0.03} \approx 11.9 \, \text{minutes}.$

Final Answer:

Juan should leave the water in the cooler for approximately 11.9 minutes.

Would you like further clarification on this solution or additional steps? Here are some related questions:

1. How is the natural logarithm used in exponential equations?
2. Why is it important not to round intermediate computations in this problem?
3. Can you explain the relationship between logarithms and exponential decay?
4. What is the significance of the constant $-0.03$ in the given function?
5. How does the cooling process change if the initial temperature is higher?

Tip: When solving equations with exponents, isolating the exponential term first simplifies the process significantly.